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College Graduates In Example 3.1 on page 197, we see that \(27.5 \%\) of US adults are college graduates. (a) Use StatKey or other technology to generate a sampling distribution for the sample proportion of college graduates using a sample size of \(n=50 .\) Generate at least 1000 sample proportions. Give the shape and center of the sampling distribution and give the standard error. (b) Repeat part (a) using a sample size of \(n=500\).

Performers in the Rock and Roll Hall of Fame From its founding through \(2015,\) the Rock and Roll Hall of Fame has inducted 303 groups or individuals, and 206 of the inductees have been performers while the rest have been related to the world of music in some way other than as a performer. The full dataset is available in RockandRoll. (a) What proportion of inductees have been performers? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion who were performers for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.38 tells us that 206 of the 303 inductees to the Rock and Roll Hall of Fame have been performers. The data are given in RockandRoll. Using all inductees as your population: (a) Use StatKey or other technology to take many random samples of size \(n=10\) and compute the sample proportion that are performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of \(p=0.68 ?\) How far away is it? (b) Repeat part (a) using samples of size \(n=20\). (c) Repeat part (a) using samples of size \(n=50\). (d) Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. A random sample of registered voters in the US is used to estimate the proportion of all US registered voters who voted in the last election.

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. Random samples of organic eggs and eggs that are not organic are used to estimate the difference in mean protein level between the two types of eggs.

SKILL BUILDER 2 In Exercises 3.45 to 3.48 , construct an interval giving a range of plausible values for the given parameter using the given sample statistic and margin of error. For \(p,\) using \(\hat{p}=0.37\) with margin of error 0.02 .

In Exercises 3.49 and 3.50 , a \(95 \%\) confidence interval is given, followed by possible values of the population parameter. Indicate which of the values are plausible values for the parameter and which are not. A \(95 \%\) confidence interval for a mean is 112.1 to \(128.2 .\) Is the value given a plausible value of \(\mu ?\) (a) \(\mu=121\) (b) \(\mu=113.4\) (c) \(\mu=105.3\)

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ \bar{x}=55 \text { and the standard error is } 1.5 . $$

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ r=0.34 \text { and the standard error is } 0.02 \text { . } $$

Adolescent Brains Are Different Researchers continue to find evidence that brains of adolescents behave quite differently than either brains of adults or brains of children. In particular, adolescents seem to hold on more strongly to fear associations than either children or adults, suggesting that frightening connections made during the teen years are particularly hard to unlearn. In one study, \({ }^{25}\) participants first learned to associate fear with a particular sound. In the second part of the study, participants heard the sound without the fear-causing mechanism, and their ability to "unlearn" the connection was measured. A physiological measure of fear was used, and larger numbers indicate less fear. We are estimating the difference in mean response between adults and teenagers. The mean response for adults in the study was 0.225 and the mean response for teenagers in the study was \(0.059 .\) We are told that the standard error of the estimate is 0.091 . (a) Give notation for the quantity being estimated. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated. (d) Is this an experiment or an observational study?